Possible definitions of exponential function

The exponential function is the unique solution of the initial value problem

$y'(x)=y(x) , \quad y(0)=1$.

We can also define $e^x$ as follows:

• the inverse function of $\ln x$, defining $\ln x$ independently as follows

$$\ln x := \int_1^x \frac{dt}{t}$$

• the unique solution to IVP $f'(x)=f(x)$ with $f(0)=1$ which existence is guaranteed by Existence and Uniqueness” Theorems for first order IVP

Define the value at rationals via powers and roots and then show that there is a unique continuous function which agrees with these values.

First define it for the natural numbers:

Define $e^2 = e \times e$, $e^3 = e \times e \times e $, etc.

Now define it for other integers:

$e^0 = 1$, $e^{-n} = \frac{1}{e^n}$, etc.

Now for other rational numbers (getting a bit harder):

$e^{\frac{p}{q}} = \sqrt[q]{e^p}$

Finally for irrational numbers $x$, you will need to prove that this definition evaluated for any sequence of rational numbers which converge to $x$ has a limit and that it is the same for all sequences which converge to $x$.

This is hard, especially the last step, but I think that it fits a common naive idea of what exponentiation is. We usually learn it in this sequence.